Which of the following best describes the graph of any quadratic function $y=a{x}^2+bx+c$, with $a\ne 0$?

For the function $y=-\frac{1}{2}{(x+3)}^2+2$, what is the vertex and the equation of the axis of symmetry?

Solve $2(x-4{)}^2-18=0$ to find the $x$-intercepts of the parabola.

Use the axis-of-symmetry formula $x_v=-\frac{b}{2a}$ to find the vertex of $y=-x^2+4x+1$.

A toy rocket's vertical position, in units of meters, is modeled by: $h\left(t\right)=-5t^2+20t+1$, where $t$ is in seconds. If aerodynamic drag doubles the magnitude of the $t^2$ coefficient, such that the model becomes:$h_2\left(t\right)=-10t^2+20t+1$, determine how the time where the rocket will reach its maximum height and how the maximum height itself will change as a result.